Inflation, a period of accelerated expansion in the early evolution of the Universe, provides an elegant solution to the horizon and flatness problem, and at the same time a mechanism to generate the primordial seeds required to explain the observed large scale structure [1]}, [2]}, [3]}. Such a period can be easily modelled by a dynamical scalar field, the inflaton, with the appropriate potential and interactions. According to the most recent CMB data [4]}, a successful inflationary model should predict a quasi-adiabatic and gaussian primordial spectrum of perturbations, with spectral index \(n_s=0.965 \pm 0.004\) , and less than a 10% of primordial gravity waves (GW), i.e, a tensor-to-scalar ratio below \(r < 0.07\) . However, although inflation should last at least around 50-60 efolds in order to explain the horizon and flatness problem, CMB constraints only apply to roughly the first 10 e-folds of those, when the larger observable scales \(k \simeq 10^{-3}-1\)  Mpc\(^{-1}\)  leave the horizon, leaving the remaining inflationary dynamics and the primordial spectrum largely unconstrained. At smaller scales, the spectral index can turn from red-tilted to blue-tilted, and the amplitude of the primordial spectrum can be much larger than the CMB value \(P_{\cal R} = 2.1\times 10^{-9}\) . If the amplitude reaches a critical value \(P_{\cal R} \sim 10^{-2}\)  this could lead to the formation of Primordial Black Holes (PBHs) on re-entry, due to the collapse of the overdensities [5]}, [6]}, [7]}, [8]}, with a very rich phenomenology. For example non-evaporating PBHs within the appropriate mass range \(M_{PBH} > 10^{15}\)  g could be all or part of the Dark Matter content of our Universe [9]}, [10]}, [11]}, [12]}; and they can also act as a source of GW with a characteristic frequency spectrum [13]}, [14]}, [15]}. Similarly, large scalar perturbations are a source of tensor perturbations at second order, and therefore the same mechanism that leads to PBHs during inflation will lead to a larger amplitude of primordial GW on smaller scales [16]}, [17]}, [18]}.
